US-EGMO-TST 2026 Problem 6

Let $ABC$ be an acute scalene triangle with circumcircle $\Gamma$, and let $M$ the midpoint of $BC$. Let $\omega$ be the circumcircle of triangle formed by $BC$ and the two common external tangents of the circumcircles $ABM$ and $ACM$. Prove that the internal bisector of $\angle BAC$ and the perpendicular bisector of $AM$ intersect on the radical axis of $\omega$ and $\Gamma$.